Real and Imaginary Parts of Decidability-Making Gilbert Giacomoni
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[Math.LO] 2 Mar 2011 39,18B40
GROUPOIDS, IMAGINARIES AND INTERNAL COVERS EHUD HRUSHOVSKI Abstract. Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T . We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between: covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of T σ, and definable groupoids. As a corollary, we describe the imaginary elements of families of finite- dimensional vector spaces over pseudo-finite fields. The questions this manuscript addresses arose in the course of an investigation of the imag- inary sorts in ultraproducts of p-adic fields. These were shown to be understandable given the imaginary sorts of certain finite-dimensional vector spaces over the residue field. The residue field is pseudo-finite, and the imaginary elements there were previously studied, and shown in fact to be eliminable over an appropriate base. It remains therefore to describe the imaginaries of finite-dimensional vector spaces over a field F , given those of F . I expected this step to be rather easy; but it turned out to become easy only after a number of issues, of interest in themselves, are made clear. Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T . Internal covers were recognized as central in the study of totally categorical structures, but nevertheless remained mysterious; it was not clear how to describe the possible T ′ from the point of view of T . We give an account of this here, in terms of groupoids in place of equivalence relations. -
Oort Conjecture for the Moduli Space of Abelian Surfaces
COMPOSITIO MATHEMATICA The Andr´e{Oortconjecture for the moduli space of abelian surfaces Jonathan Pila and Jacob Tsimerman Compositio Math. 149 (2013), 204{216. doi:10.1112/S0010437X12000589 FOUNDATION COMPOSITIO MATHEMATICA Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.234, on 28 Sep 2021 at 16:32:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X12000589 Compositio Math. 149 (2013) 204{216 doi:10.1112/S0010437X12000589 The Andr´e{Oortconjecture for the moduli space of abelian surfaces Jonathan Pila and Jacob Tsimerman Abstract We provide an unconditional proof of the Andr´e{Oortconjecture for the coarse moduli space A2;1 of principally polarized abelian surfaces, following the strategy outlined by Pila{Zannier. 1. Introduction and notation Let Ag;1 denote the coarse moduli space of principally polarized abelian varieties of dimension g. Our main theorem is the following, proving the Andr´e{Oortconjecture for A2;1. Theorem 1.1. Let V ⊂ A2;1 be an algebraic subvariety, which is equal to the Zariski closure of its CM points. Then V is a special subvariety. Here, a subvariety means a relatively closed irreducible subvariety. Varieties will be identified with their sets of complex-valued points. We follow the general strategy of Pila{Zannier. Set Hg to be the Siegel upper half space t Hg = fZ 2 Mg×g(C) j Z = Z ; Im(Z) > 0g: We denote by π : Hg !Ag;1 t g(g+1)=2 the natural projection map. The set fZ 2 Mg×g(C)jZ = Z g is naturally identified with C , g(g+1)=2 g(g+1) identifying Hg with an open domain. -
1 1. Stability Stability Is Fundamentally a Local Hypothesis, Indeed A
1 1 1. Stability Stability is fundamentally a local hypothesis, indeed a combinatorial hy- pothesis on a bipartite graph. A bi-partite graph is just a triple of sets G = (G1;G2; I) with I G1 G2; one thinks of I as relating elements of ⊂ × G1 to elements of G2. An example is Ln, where G1 and G2 are two copies of 1; : : : ; n , and I is the graph of the order relation i < j. f g Definition 1.1. A bi-partite graph is stable if for some n, G does not embed the graph Ln Definition 1.2. A Γ-formula a formula equivalent to a quantifier-free formula, possibly with additional parameters. A set C is Γ-algebraically-closed if for every Γ-formula E over C defining an equivalence relation with finitely many classes, each class of E is coded by an element of C.A Γ-type over C is a maximal consistent collection of Γ-formulas over C.A Γ-type p over Γ is clearly determined by its restriction to atomic formulas; we will say p is definable if there exists a formula α(y) such that xIy p iff α(y) (or dually; we will write 2 xIy or yIx interchangeably.) If p is definable, the defining formula is unique up to logical equivalence, and will be denoted (dpx)(xIy). Atomic types over Γ are assumed consistent with Th(Γ) (equivalently, with the universal theory of Γ.) Lemma 1.3. The following are equivalent: i) Γ is stable ii) For every countable Γ0 satisfying the universal theory T of Γ, there are 8 countably many atomic types over Γ0 consistent with the universal theory of Γ0. -
Tractatus Logico-Philosophicus</Em>
University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 8-6-2008 Three Wittgensteins: Interpreting the Tractatus Logico-Philosophicus Thomas J. Brommage Jr. University of South Florida Follow this and additional works at: https://scholarcommons.usf.edu/etd Part of the American Studies Commons Scholar Commons Citation Brommage, Thomas J. Jr., "Three Wittgensteins: Interpreting the Tractatus Logico-Philosophicus" (2008). Graduate Theses and Dissertations. https://scholarcommons.usf.edu/etd/149 This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Three Wittgensteins: Interpreting the Tractatus Logico-Philosophicus by Thomas J. Brommage, Jr. A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Philosophy College of Arts and Sciences University of South Florida Co-Major Professor: Kwasi Wiredu, B.Phil. Co-Major Professor: Stephen P. Turner, Ph.D. Charles B. Guignon, Ph.D. Richard N. Manning, J. D., Ph.D. Joanne B. Waugh, Ph.D. Date of Approval: August 6, 2008 Keywords: Wittgenstein, Tractatus Logico-Philosophicus, logical empiricism, resolute reading, metaphysics © Copyright 2008 , Thomas J. Brommage, Jr. Acknowledgments There are many people whom have helped me along the way. My most prominent debts include Ray Langely, Billy Joe Lucas, and Mary T. Clark, who trained me in philosophy at Manhattanville College; and also to Joanne Waugh, Stephen Turner, Kwasi Wiredu and Cahrles Guignon, all of whom have nurtured my love for the philosophy of language. -
First Order Logic
Artificial Intelligence CS 165A Mar 10, 2020 Instructor: Prof. Yu-Xiang Wang ® First Order Logic 1 Recap: KB Agents True sentences TELL Knowledge Base Domain specific content; facts ASK Inference engine Domain independent algorithms; can deduce new facts from the KB • Need a formal logic system to work • Need a data structure to represent known facts • Need an algorithm to answer ASK questions 2 Recap: syntax and semantics • Two components of a logic system • Syntax --- How to construct sentences – The symbols – The operators that connect symbols together – A precedence ordering • Semantics --- Rules the assignment of sentences to truth – For every possible worlds (or “models” in logic jargon) – The truth table is a semantics 3 Recap: Entailment Sentences Sentence ENTAILS Representation Semantics Semantics World Facts Fact FOLLOWS A is entailed by B, if A is true in all possiBle worlds consistent with B under the semantics. 4 Recap: Inference procedure • Inference procedure – Rules (algorithms) that we apply (often recursively) to derive truth from other truth. – Could be specific to a particular set of semantics, a particular realization of the world. • Soundness and completeness of an inference procedure – Soundness: All truth discovered are valid. – Completeness: All truth that are entailed can be discovered. 5 Recap: Propositional Logic • Syntax:Syntax – True, false, propositional symbols – ( ) , ¬ (not), Ù (and), Ú (or), Þ (implies), Û (equivalent) • Semantics: – Assigning values to the variables. Each row is a “model”. • Inference rules: – Modus Pronens etc. Most important: Resolution 6 Recap: Propositional logic agent • Representation: Conjunctive Normal Forms – Represent them in a data structure: a list, a heap, a tree? – Efficient TELL operation • Inference: Solve ASK question – Use “Resolution” only on CNFs is Sound and Complete. -
3 Fundamental Class 20 3.1 Bottom Fundamental Class
JAIST Repository https://dspace.jaist.ac.jp/ Title A Categorical Description of Relativization Author(s) 吉村, 和人 Citation Issue Date 2013-03 Type Thesis or Dissertation Text version author URL http://hdl.handle.net/10119/11296 Rights Description Supervisor:石原 哉, 情報科学研究科, 修士 Japan Advanced Institute of Science and Technology A Categorical Description of Relativization By Kazuto Yoshimura A thesis submitted to School of Information Science, Japan Advanced Institute of Science and Technology, in partial fulfillment of the requirements for the degree of Master of Information Science Graduate Program in Information Science Written under the direction of Hajime Ishihara March 2013 A Categorical Description of Relativization By Kazuto Yoshimura (1110069) A thesis submitted to School of Information Science, Japan Advanced Institute of Science and Technology, in partial fulfillment of the requirements for the degree of Master of Information Science Graduate Program in Information Science Written under the direction of Hajime Ishihara and approved by Hajime Ishihara Kazuhiro Ogata Mizuhito Ogawa February 2013 (Submitted) Copyright c 2013 by Kazuto Yoshimura Contents 1 Introduction 2 2 Preliminairies 5 2.1 Set Theoretic Notations . 5 2.2 Category . 9 2.3 Functor . 18 3 Fundamental Class 20 3.1 Bottom Fundamental Class . 20 3.1.1 Classes of Morphisms . 20 3.1.2 Factorization System . 22 3.1.3 Pre-Effectiveness . 27 3.2 Imaginary Fundamental Class . 29 3.2.1 Extending Fundamental classes . 29 3.2.2 Imaginary Operator . 32 3.2.3 The Strongest Extension Ability . 34 3.2.4 Dense morphisms . 37 3.3 Limit Fundamental Class . 38 3.3.1 Effectiveness . -
The André-Oort Conjecture for the Moduli Space Of
THE ANDRE-OORT´ CONJECTURE FOR THE MODULI SPACE OF ABELIAN SURFACES JONATHAN PILA AND JACOB TSIMERMAN Abstract. We provide an unconditional proof of the Andr´e-Oortcon- jecture for the coarse moduli space A2;1 of principally polarized Abelian surfaces, following the strategy outlined by Pila-Zannier. 1. introduction and notation Set Hg to be the Siegel upper half space t Hg = fZ 2 Mg(C) j Z = Z ; Im(Z) > 0g: Let Ag;1 denote the coarse moduli space of principally polarized Abelian varieties of dimension g. Our main theorem is the following, proving the Andr´e-Oortconjecture for A2;1 : Theorem 1.1. Let V ⊂ A2;1 be an algebraic subvariety, which is equal to the Zariski closure of its CM points. Then V is a special subvariety. We follow the general strategy of Pila-Zannier. One ingredient we need is the following Ax-Lindemann-Weierstrass theorem: We denote by π : H2 !A2;1 6 the natural projection map. We consider H2 ⊂ R with coordinates provided 3 by real and imaginary part of the complex coordinates in C , and call a set 6 semialgebraic if it is a semialgebraic set in R . Let V ⊂ A2;1 be an algebraic variety, Z = π−1(V ), and Y ⊂ Z an irreducible semialgebraic subvariety 0 of H2. We say that Y is maximal if for all semialgebraic subvarieties Y containing Y with Y 0 ⊂ Z, Y is a component of Y 0. Theorem 1.2. Let Y ⊂ Z be a maximal semialgebraic variety. Then Y is a weakly special subvariety. Fix Fg ⊂ Hg to be the standard fundamental domain [8]. -
View of first-Order Logic and O-Minimality
TAMENESS RESULTS FOR EXPANSIONS OF THE REAL FIELD BY GROUPS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Michael A. Tychonievich, B.S., M.S. Graduate Program in Mathematics The Ohio State University 2013 Dissertation Committee: Prof. Christopher Miller, Advisor Prof. Timothy Carlson Prof. Ovidiu Costin Prof. Daniel Verdier c Copyright by Michael A. Tychonievich 2013 ABSTRACT Expanding on the ideas of o-minimality, we study three kinds of expansions of the real field and discuss certain tameness properties that they possess or lack. In Chapter 1, we introduce some basic logical concepts and theorems of o-minimality. In Chapter 2, we prove that the ring of integers is definable in the expansion of the real field by an infinite convex subset of a finite-rank additive subgroup of the reals. We give a few applications of this result. The main theorem of Chapter 3 is a struc- ture theorem for expansions of the real field by families of restricted complex power functions. We apply it to classify expansions of the real field by families of locally closed trajectories of linear vector fields. Chapter 4 deals with polynomially bounded o-minimal structures over the real field expanded by multiplicative subgroups of the reals. The main result is that any nonempty, bounded, definable d-dimensional sub- manifold has finite d-dimensional Hausdorff measure if and only if the dimension of its frontier is less than d. ii To my father, who has fallen asleep. iii ACKNOWLEDGMENTS As a graduate student, I owe thanks to many people, but I will only list a few here. -
Frege and the Logic of Sense and Reference
FREGE AND THE LOGIC OF SENSE AND REFERENCE Kevin C. Klement Routledge New York & London Published in 2002 by Routledge 29 West 35th Street New York, NY 10001 Published in Great Britain by Routledge 11 New Fetter Lane London EC4P 4EE Routledge is an imprint of the Taylor & Francis Group Printed in the United States of America on acid-free paper. Copyright © 2002 by Kevin C. Klement All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any infomration storage or retrieval system, without permission in writing from the publisher. 10 9 8 7 6 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data Klement, Kevin C., 1974– Frege and the logic of sense and reference / by Kevin Klement. p. cm — (Studies in philosophy) Includes bibliographical references and index ISBN 0-415-93790-6 1. Frege, Gottlob, 1848–1925. 2. Sense (Philosophy) 3. Reference (Philosophy) I. Title II. Studies in philosophy (New York, N. Y.) B3245.F24 K54 2001 12'.68'092—dc21 2001048169 Contents Page Preface ix Abbreviations xiii 1. The Need for a Logical Calculus for the Theory of Sinn and Bedeutung 3 Introduction 3 Frege’s Project: Logicism and the Notion of Begriffsschrift 4 The Theory of Sinn and Bedeutung 8 The Limitations of the Begriffsschrift 14 Filling the Gap 21 2. The Logic of the Grundgesetze 25 Logical Language and the Content of Logic 25 Functionality and Predication 28 Quantifiers and Gothic Letters 32 Roman Letters: An Alternative Notation for Generality 38 Value-Ranges and Extensions of Concepts 42 The Syntactic Rules of the Begriffsschrift 44 The Axiomatization of Frege’s System 49 Responses to the Paradox 56 v vi Contents 3. -
Toward a Logic of Everyday Reasoning
Toward a Logic of Everyday Reasoning Pei Wang Abstract: Logic should return its focus to valid reasoning in real-world situations. Since classical logic only covers valid reasoning in a highly idealized situation, there is a demand for a new logic for everyday reasoning that is based on more realistic assumptions, while still keeps the general, formal, and normative nature of logic. NAL (Non-Axiomatic Logic) is built for this purpose, which is based on the assumption that the reasoner has insufficient knowledge and resources with respect to the reasoning tasks to be carried out. In this situation, the notion of validity has to be re-established, and the grammar rules and inference rules of the logic need to be designed accordingly. Consequently, NAL has features very different from classical logic and other non-classical logics, and it provides a coherent solution to many problems in logic, artificial intelligence, and cognitive science. Keyword: non-classical logic, uncertainty, openness, relevance, validity 1 Logic and Everyday Reasoning 1.1 The historical changes of logic In a broad sense, the study of logic is concerned with the principles and forms of valid reasoning, inference, and argument in various situations. The first dominating paradigm in logic is Aristotle’s Syllogistic [Aristotle, 1882], now usually referred to as traditional logic. This study was carried by philosophers and logicians including Descartes, Locke, Leibniz, Kant, Boole, Peirce, Mill, and many others [Bochenski,´ 1970, Haack, 1978, Kneale and Kneale, 1962]. In this tra- dition, the focus of the study is to identify and to specify the forms of valid reasoning Pei Wang Department of Computer and Information Sciences, Temple University, Philadelphia, USA e-mail: [email protected] 1 2 Pei Wang in general, that is, the rules of logic should be applicable to all domains and situa- tions where reasoning happens, as “laws of thought”. -
Formalizing Common Sense Reasoning for Scalable Inconsistency-Robust Information Integration Using Direct Logictm Reasoning and the Actor Model
Formalizing common sense reasoning for scalable inconsistency-robust information integration using Direct LogicTM Reasoning and the Actor Model Carl Hewitt. http://carlhewitt.info This paper is dedicated to Alonzo Church, Stanisław Jaśkowski, John McCarthy and Ludwig Wittgenstein. ABSTRACT People use common sense in their interactions with large software systems. This common sense needs to be formalized so that it can be used by computer systems. Unfortunately, previous formalizations have been inadequate. For example, because contemporary large software systems are pervasively inconsistent, it is not safe to reason about them using classical logic. Our goal is to develop a standard foundation for reasoning in large-scale Internet applications (including sense making for natural language) by addressing the following issues: inconsistency robustness, classical contrapositive inference bug, and direct argumentation. Inconsistency Robust Direct Logic is a minimal fix to Classical Logic without the rule of Classical Proof by Contradiction [i.e., (Ψ├ (¬))├¬Ψ], the addition of which transforms Inconsistency Robust Direct Logic into Classical Logic. Inconsistency Robust Direct Logic makes the following contributions over previous work: Direct Inference Direct Argumentation (argumentation directly expressed) Inconsistency-robust Natural Deduction that doesn’t require artifices such as indices (labels) on propositions or restrictions on reiteration Intuitive inferences hold including the following: . Propositional equivalences (except absorption) including Double Negation and De Morgan . -Elimination (Disjunctive Syllogism), i.e., ¬Φ, (ΦΨ)├T Ψ . Reasoning by disjunctive cases, i.e., (), (├T ), (├T Ω)├T Ω . Contrapositive for implication i.e., Ψ⇒ if and only if ¬⇒¬Ψ . Soundness, i.e., ├T ((├T) ⇒ ) . Inconsistency Robust Proof by Contradiction, i.e., ├T (Ψ⇒ (¬))⇒¬Ψ 1 A fundamental goal of Inconsistency Robust Direct Logic is to effectively reason about large amounts of pervasively inconsistent information using computer information systems. -
How Peircean Was the “'Fregean' Revolution” in Logic?
HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? Irving H. Anellis Peirce Edition, Institute for American Thought Indiana University – Purdue University at Indianapolis Indianapolis, IN, USA [email protected] Abstract. The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912–1986) and Don- ald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848–1925) in his Begriffsschrift of 1879, and with Bertrand Rus- sell (1872–1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806–1871), George Boole (1815–1864), Charles Sanders Peirce (1838–1914), and Ernst Schröder (1841–1902) as belonging to the Aristotelian tradi- tion. The “Booleans” are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege’s innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort’s list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another’s logical writings. AMS (MOS) 2010 subject classifications: Primary: 03-03, 03A05, 03C05, 03C10, 03G27, 01A55; secondary: 03B05, 03B10, 03E30, 08A20; Key words and phrases: Peirce, abstract algebraic logic; propositional logic; first-order logic; quantifier elimina- tion, equational classes, relational systems §0.